The structure of interval orders with no infinite antichain

TL;DR

This paper characterizes the structure of interval orders without infinite antichains, showing they decompose into simpler components and are countable and scattered, with examples illustrating their complexity.

Contribution

It proves that such interval orders have a Gallai decomposition and are at most countable and scattered, providing new structural insights.

Findings

Every such interval order has a Gallai decomposition.

Prime interval orders without infinite antichains are at most countable and scattered.

Constructs examples with chain of maximal antichains of arbitrary Hausdorff rank.

Abstract

We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhomm\'e (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $\alpha$ we exhibit an example of a well-quasi-ordered prime interval order $P_\alpha$ whose chain of maximal antichains has Hausdorff rank $\alpha$.